Problem of the Month (November 2006)

This month we consider two problems about tiling polyominoes.

1. Given two polyominoes P1 and P2 with the same area, what is the smallest area of a polyomino S so that one copy of P1 and S can tile the same shape as one copy of P2 and S? Most of the answers for pentominoes are small, but some of the answers are surprisingly large. Can you find the polyominoes S for the largest cases? Can you find a polyomino S that works for the unknown case? What are the largest answers for hexominoes and larger polyominoes? What about other polyforms, like polyhexes and polyiamonds?

2. Given two polyominoes P1 and P2 with the same area, and a positive integer n, what is the smallest shape S that can be tiled by P1 in exactly n ways, and can be tiled by P2 in exactly n ways? Can you find smaller solutions for the triominoes? What are best results for larger n? What about the pairs of tetrominoes? What about other polyforms, like polyhexes and polyiamonds? What if we want S that can be tiled in n ways by P1, P2, or P3?

ANSWERS

1. The polyomino addition solutions for tetrominoes are fairly trivial:

Tetromino Solutions

1

1

2

2

1

1

1

1

1

1

The solutions for pentominoes are more interesting. There are several cases that require area 5 or more, one really huge solution, and one unsolved case.

Pentomino Solutions

2

5

1

1

2

2

4

2

5

5

5

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

23

1

1

1

2

1

1

1

7

1

1

2

1

1

1

1

1

1

1

1

1

?

1

1

1

1

5

4

1

1

5

1

1

1

1

1

5

5

7

23

(Corey Plover)

In 2012, George Sicherman studied the problem for triples of pentominoes, finding these results:

The largest solutions for hexominoes are below.

Large Hexomino Solutions

6

7

8

9

(George Sicherman)

11

(George Sicherman)

12+

(George Sicherman)

Here are the unsolved cases for hexomino pairs.

Here are some large cases for heptomino pairs:

Corey Plover is not a big fan of flipping polyominoes over, so he was interested in the solutions with this restriction.

Non-Flip Pentomino Solutions

2

2

7

1

1

1

1

2

2

2

4

2

?

?

5

5

5

1

1

2

1

1

1

1

2

1

1

1

1

1

2

1

1

1

1

2

1

1

2

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

2

1

1

1

1

1

23

1

1

1

2

1

1

2

1

2

1

1

2

?

1

1

1

2

1

2

1

1

2

2

1

?

1

1

2

1

2

1

1

1

2

1

1

2

1

1

2

1

1

1

1

2

1

2

1

1

1

1

1

2

1

?

1

1

1

1

1

1

2

?

1

1

1

1

1

1

23

?

1

1

1

1

23

1

1

1

1

1

1

2

1

1

1

2

1

2

5

5

Here are the largest known solutions:

(George Sicherman)

(George Sicherman)

(George Sicherman)

Corey Plover noticed that there are smaller solutions if we allow king-connected polyominoes or disconnected polyominoes: